How to simulate Miura-fold
I was able to create that animation just using shape keys.


Start out with a plane that has the same geometry as the folds in a Miura fold.

Add two shape keys on that mesh. The first will be the "Basis," the second will be the one in which you edit the mesh. (I used more then two, but two will work fine.)

Tab in to Edit mode and start moving the edges together. The only tricky part is keeping the edges the same length. You will need to turn on Length in the Mesh Display heading of the properties region. (I duplicated the unedited mesh so I could check the edge lengths.)



Start by selecting every second edge loop and moving it up along the Z a bit less then the length of the edges. (In my mesh the shortest edge was about 2.66, so I moved the edges up 2.5.)
Starting at the end, select each edge loop and move it along the X until the lengths match the original length. (It does not have to be perfect, in the gif I'm sure not.) Continue moving the edges closer together until you are done with that axis.
Simply do the same thing for the other axis. Starting at the top (or bottom) select each edge loop and move it along the Y axis until it is the same length as the original.



In this gif I only do three edges, just because it is all the same.

4. Add a bevel and a Subsurf modifier to the mesh. You will need to crease the outside edges, to keep the corners from rounding.